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Full text of "Mathematical And Physical Papers - Iii"

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beginning to the arc at the end of an oscillation, we must put tf = r in (168), whence, neglecting the effect of the wire, we obtain
If now A&' be the correction to be applied to kf in this formula on account of the wire, since k', &/ are combined together in the expression for the arc just as k, kl in the expression for the time, we get
A&'=A'A   ..................... (170),
and the approximate formulae (115) give
AL A# = -  Aft ..................... (171),
whence the numerical value of A// is easily deduced from that of A/, which has been already calculated.    We get also from (52) fc^fc-i + Kfc-i)* .................. (172),
whence k' may be readily deduced from k, which has been already calculated.
74. Before comparing these formulae with BessePs experiments, it will be proper to enquire how far the latter are satisfied by supposing the arcs of oscillation to decrease in geometric progression. In Bessel's tables the arc is registered in the column headed //,. This letter denotes the number of French lines read off on a scale placed behind the wire, and a little above the sphere, and is reckoned from the position of instantaneous rest of the wire on one side of the vertical to the corresponding position on the other side. The distance of the scale from the axis of suspension being given, as well as the correction to be applied to \L on account of parallax, the arc of oscillation may be readily deduced. However, for our present purpose, any quantity to which the arc is proportional will do as well as the arc itself, and /I,, though strictly proportional to the tangent of the arc, may be regarded as proportional to the arc itself, inasmuch as the initial arc usually amounted to only about 50' on each side of the vertical.
Now we may form a very good judgment as to the degree of accuracy of the geometric formula by comparing the arc observed